Monochromatic K_r-decompositions of graphs

Given graphs G and H, and a coloring of the edges of G with k colors, a monochromatic H-decomposition of G is a partition of the edge set of G such that each part is either a single edge or forms a monochromatic graph isomorphic to H. Let φ_k(n,H) be the smallest number φ such that any graph G of order n and any coloring of its edges with k colors, admits a monochromatic H-decomposition with at most φ parts. Here, we study the function φ_k(n, K_r) for k ≥ 2 and r ≥ 3.